Question

solve the equation; 3a/4-2a/3=7/4

Answer 1

Answer:  " a = 21 " .

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Step-by-step explanation:

Given:  3a/4-2a/3 = 7/4 ;  Solve for "a" ;

Rewrite as: (3a/4) - (2a/3) = (7/4) ;

Now, for each of the three (3) "denominator values" in "fraction form" within the equation given:

      → Find the "LCD" ["Least Common Denominator"]:
The denominators are:  4, 3, and 4 ;  

 that is:  "3" and "4" ;
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To find the LCD: First; multiply the denominators:  "4 * 3 = 12" .
 So; the value "12" could be the LCD;  so, the value for the LCD is no greater than "12" ; however, there could be a smaller value.
To determine the LCD:
List the multiples of the given denominators:
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3: 3, 6, 9, 12, 15 .... ;
4: 4, 8, 12, 16... ;
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We find that "12" is, in fact, the LCD of "3" and 4:
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We can multiply each side of the equation by "12" ; to eliminate the "fractional values" :
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   $12*[\frac{3a}{4} - \frac{2a}{3}] = 12*[{\frac{7}{4}]$
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Note the "distributive property" of multiplication:
  →  a(b+c) = ab + ac ;

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As such:
$12*[\frac{3a}{4} - \frac{2a}{3}] = 12*[{\frac{7}{4}]$ ;

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Let us start with the "left-hand side" of the equation:
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$12*[\frac{3a}{4} - \frac{2a}{3}]$ ;

 =  $[12*\frac{3a}{4}] + [-12 * \frac{2a}{3}]$ ;
 =  $[12*\frac{3a}{4}] - [12 * \frac{2a}{3}] ;$
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Note:  "  $[12*\frac{3a}{4}]$ " ;

                     $= \frac{12}{1} * \frac{3a}{4}$ ;

       →  The "12" cancels to a "3" ; and the "4" cancels to a "1" ;
 since: "12÷4 = 3" ; and since:  "4÷4 = 1" ;
       → and we can rewrite the "left-hand-side" expression as:
       →  "   $\frac{3}{1} * \frac{3a}{1}$ " ;  which we can simplify as:  

               →  "3 * 3a" ; which we can simplify as:  " 9a " .
then we have:  " [12 * $\frac{2a}{3}$ ] " ;

 which equals:
 =   " $\frac{12}{1} * \frac{2a}{3}$ " ;
Note: The "12" cancels out to a "4"; & the "3" cancels out to a "1" ;

  →  {since:  "(12 ÷ 3 = 4)"; & since: "(3 ÷ 3 = 1)" ;
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→ And we can rewrite the expression as:
     →  " $\frac{4}{1} *\frac{2a}{1}$ " ;  which we can simplify as:
     →  " 4 * 2a " ; which we can simplify/calculation as:  " 8a " ;
Now, we can rewrite the expression of the "left-hand side"
of the equation as:
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    →  " [9a] − [8a] " ; (don't forget to carry down the "minus sign"!) ;
which we can simplify/calculate to get:
  →  " [9a − 8a] " ;  which we can further simplify/calculate;

  →  to get:
        → " 1a " ;  or:  "a" —the value for which we wish to solve!
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Now, let us examine the "right-hand side" of the equation:
 ____

→  " $\frac{12}{1} * \frac{7}{4}$ " ;
Note:  The "12" cancels out to a "3" ; & the "4" cancels out to a "1" ;
       → {Since:  "12 ÷4 = 3 " ;  &  since:  "4 ÷ 4 = 1 "} ;

And we can rewrite the expression as:  
    →  " $\frac{3}{1} * \frac{7}{1}$ " ; which we can simplify as:
           → " 3 * 7 " ;  which can simplify/calculate to get:  " 21" ;
⇒  Now, let us rewrite the equation; by using our simplified values for both the "left-hand side" and the "right-hand side" of the equation; to solve for "a" :
 ⇒  a = 21 ;  

→  which is the correct answer:  
        → " a = 21 " .
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  Hope this helps!

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Answer 2

EXPLANATION:

Given equation is {(3a)/4} - {(2a)/3} =7/4

Write all the numerators with one common denominator by converting them into like fractions.

Take the LCM of the denominator on LHS i.e., 3 and 4 is 12

⇛{(3a*3 - 2a*4)/12} = 7/4

⇛{(9a - 8a)/12} = 7/4

⇛{(1a)/12} = 7/4

⇛a/12 = 7/4

Since, (a/b) = (c/d) ⇛a(d) = b(v) ⇛ad = bc

Where, a = 1, b = 12, c = 7 and d = 4

On, applying cross multiplication then

⇛a(4) = 7(12)

⇛4a = 84

Shift the number four (4) from LHS to RHS, changing it's sign.

⇛a = 84/4

Simplify the fraction on RHS.

⇛a = {(84÷2)/(4÷2)}

⇛a = 42/2

⇛a = {(42÷2)/(2÷2)}

⇛a = 21/1

Therefore, a = 21

Answer: The Value of a for the given problem is 21.

VERIFICATION:

Check, whether the value of x is true or false.

If x = 21 then LHS of the equation is

{(3a)/4} - {(2a)/3} =7/4

On substituting the value of x in equation then

⇛[{3(21)}/4] - [{2(21)}/3] = 7/4

⇛(63/4) - (42/3) = 7/4

Write all the numerators with one common denominator by converting them into like fractions.

Take the LCM of the denominator on LHS i.e., 3 and 4 is 12.

⇛{(63*3 - 42*4)/12} = 7/4

⇛{(189 - 168)/12} = 7/4

⇛21/12 = 7/4

Now, reduce the LHS fraction in lowest form by cancelling method.

⇛{(21÷3)/(12÷3)} = 7/4

⇛7/4 = 7/4

LHS = RHS is true for a = 21.

Hence, verified.

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